# Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the case of $\mathbb{R}$, it follows from distribution theory that all tempered distributions (including Dirac deltas) have Fourier transforms. I would like to know how these ideas extend to all finite Abelian groups, in particular:

Question 1: how does one define distributions over locally compact Abelian groups?

Question 2: what are the distributions that can be Fourier transformed? what would sets of distributions (for any Abelian group) would be closed under the Fourier transform of the group?

Note

I noticed the existence of these Schwartz-Bruhat functions, which seem to be related to my questions. If the theory of these is the key to the answer, a good reference would also be welcome.

On the $n$-dimensional Euclidian space $\mathbb{R}^n$, the class of tempered distributions is defined in terms of the Schwartz space $\mathcal{S} (\mathbb{R}^n)$, which is the space of all functions $f \in C^{\infty} (\mathbb{R}^n)$ such that for every pair of multi-indices $\alpha, \beta$ there exists constants $C_{\alpha, \beta} > 0$ satisfying $$\rho_{\alpha, \beta} (f) := \sup_{x \in \mathbb{R}^n} \bigg| x^{\alpha} \partial^{\beta} f(x) \bigg| \leq C_{\alpha, \beta}.$$

On $\mathcal{S} (\mathbb{R}^n)$, the Fourier transform $\mathcal{F}$ is defined as $$(\mathcal{F} f)(\xi) = \int_{\mathbb{R}^n} f(x) e^{-2\pi i x \xi} dx,$$ and one can show that $\mathcal{F} f \in \mathcal{S} (\mathbb{R}^n)$ whenever $f \in \mathcal{S}(\mathbb{R}^n)$. In fact, the Fourier transform $\mathcal{F}$ is a homeomorphism from $\mathcal{S}(\mathbb{R}^n)$ onto $\mathcal{S}(\mathbb{R}^n)$.

The class of tempered distributions on $\mathbb{R}^n$ is now defined as the topological dual space $(\mathcal{S}(\mathbb{R}^n))'$, i.e., a tempered distribution is a continuous linear functional on $\mathcal{S}(\mathbb{R}^n)$. The Fourier transform $\mathcal{F} : \mathbb{S}(\mathbb{R}^n) \to \mathbb{S}(\mathbb{R}^n)$ can be canonically extended to a mapping $\mathcal{F} : \mathbb{S}((\mathbb{R}^n))' \to (\mathbb{S}(\mathbb{R}^n))'$.

A generalisation of the Fourier transform to tempered distributions on locally compact Abelian groups $G$ is straightforward whenever there is an appropriate definition of the Schwartz space on $G$. The space $\mathcal{L} (G)$ defined next is such a generalisation:

Let $\mathcal{A}(G)$ be the space of all function $f \in L^{\infty} (G)$ for which there exists a compact set $C(f) \subset G$ such that for every $n \in \mathbb{N}$ there exists a constant $C_n$ such that for each $k \in \mathbb{N}^+$ $$\| f|_{G \setminus C(f)^k} \|_{\infty} \leq C_n k^{-n}.$$ Denote $$\mathcal{L}(G) := \{ f \in \mathcal{A} (G) \; | \; \mathcal{F} f \in \mathcal{A} (\widehat{G}) \}.$$

The space $\mathcal{L}(G)$ was introduced by Osborne in his paper On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups, and it is a characterisation of the Schwartz-Bruhat space $\mathcal{S}_b (G)$, which was introduced by Bruhat's Distributions sur un group localement compact et applications sons a l'étude de représentations des groups p-adiques. The original definition of the Schwartz-Bruhat space given by Bruhat is probably a more straightforward generalisation of $\mathcal{S}(\mathbb{R}^n)$ since it uses so-called polynomial differential operators. However, the treatment by Bruhat uses the structure theory for locally compact Abelian groups extensively, which is clearly more involved than needed in order to define $\mathcal{S}_b (\mathbb{R})$ by Osborne's characterisation.

An English text that I would recommend for a treatment of Bruhat's original definition of the Schwartz-Bruhat space is Wawrzy´nczyk's On Tempered Distributions and Bochner-Schwartz Theorem on Arbitrary Locally Compact Abelian Groups.

• It sounds interesting that the concept of tempered distribution can be extended without a notion of smoothness. I may not be able to look up the text right now, but thank you for providing such appropriate and educated information. Oct 26, 2016 at 9:42
• Thanks for the detailed answer. By the time this came, I had already found the paper of Osborne and read it. I also read the original paper of Bruhat in French. Something that I find confusing in his paper is how these so-called "polynomial differential operators" can be defined for discrete groups like $\mathbb{Z}$ or $\mathbb{Z}_2$. Do you have a reference for that? Nov 18, 2016 at 12:40